Londoners from across the city have already donated so many crates of clothes, water and food.
On Thursday, debris continued to fall from the 24-floor tower's dark skeleton as the acrid smell of burnt plastic wafted in the wind after a fire in which at least 17 people lost their lives.
Behind the security cordon at the foot of the tower, a moving show of solidarity has emerged, mixing with resentment of local residents who feel hard done by in what is Britain's richest neighbourhood.
Londoners from across the city have already donated so many crates of clothes, water and food that the local authorities have called for no further help.
"Notting Hill is the poshest part of London, but we need to come together," Sarah Archer, 34, told AFP in front of a blue ping-pong table that has become the impromptu meeting point for volunteers.
"People feel helpless, they do not know what to do, so they give," said Sonia, an Algerian woman who was helping a friend look for missing family members.
"We all know each other here, it's a very mixed neighbourhood, with Muslims and non-Muslims, many families with young children," she explained.
She described hearing screams of anguish from those trapped in the tower on the night of the disaster.
Grenfell Tower is located in Kensington and Chelsea, statistically the richest neighbourhood in the country, but with the most pronounced wealth disparities.
As with many boroughs in the British capital, crossing the road can take you from an idyllic scene of metropolitan chic to a world of concrete and economic hardship.
Former prime minister David Cameron on Wednesday tweeted his "horror" at the "terrible fire in my local neighbourhood."
The former leader lives a few minutes walk away in Notting Hill, famed for its charming houses with multicoloured facades, its annual carnival and a Hollywood film featuring Hugh Grant.
But the image projected to the world is a long way from the reality of life around Grenfell Tower and the three other 20-plus-floor blocks nearby.
"I have never really felt safe in these towers, nobody does," said Soran Karimi.
"There are so many problems."
The 31-year-old told AFP that he could never forget the horrors of what he witnessed early Wednesday morning, and like many in the neighbourhood, accuses local officials of wilful neglect.
"If the fire alarm was working, most people would be safe," he said.
"If this happened in Chelsea, it would not have taken this long to control the fire, but this part is poorer."
Prime Minister Theresa May paid a private visit to emergency services on site Thursday, but Karimi was unimpressed.
"She is not here to be in solidarity with the people, she is just here for her own name."
Nana Akuffo, 46, a chef living in one of the nearby towers, said he hoped that the incident would force a radical rethink on social housing in the city.
"Kensington and Chelsea are moving out people who grew up in the area and building plush houses for the rich people," he said. "Hopefully this is a wake-up call to them."
It has been suggested that the recently-fitted cladding on the outside of Grenfell may have accelerated the fire, a move rapper Akala said was driven by the wealth divide.